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We investigate how pulsar timing array (PTA) measurements of the nanoHertz gravitational-wave background (GWB) can constrain models for the growth history of supermassive black holes (SMBHs) and how active galactic nucleus (AGN) and stellar feedback models can affect GWB predictions. Feedback regulates supermassive black hole (SMBH) growth, altering the black hole mass function (BHMF). Using BHMFs drawn from multiple cosmological simulation suites including IllustrisTNG, MillenniumTNG, Simba, and CAMELS, and combining these with a quasar-based SMBH binary population framework, we predict the resulting GWB amplitude under a range of different stellar and AGN feedback prescriptions. We find that the choice of both stellar and AGN feedback models alters the high-mass end of the BHMF and changes the predicted GWB amplitude by up to a factor of 2 for the fiducial simulations and a factor 10 for extreme feedback variations in CAMELS. Models with inefficient or absent AGN feedback produce abundant SMBHs and yield GWB amplitudes consistent with PTA data, yet fail in producing realistic galaxies. Fiducial models of AGN and stellar feedback suppress SMBH growth too much and under-predict the expected signal, an effect which could possibly be mitigated by more realistic black hole seeding and growth prescriptions. The mismatch between the GWB amplitudes predicted by cosmological simulations and those observed by PTAs suggests that SMBH growth is more efficient or occurs earlier than captured by current models. This demonstrates that PTA measurements provide a powerful new probe of feedback physics and the SMBH population.
